It's not an issue. You just have to carry your passport when you'll cross a border (which I assume you would have done anyway).
Due to the temporary internal borders, will there be immigration/passport controls at the departure airports and at the arrival airports?
You'll most likely have to identify yourself before boarding the plane, but this has nothing to do with the internal boardborder controls. It's possible there will be a border check at the arrival airport, but actually unlikely.
Likewise, will there be passport controls at the train stations when leaving Brussels and arriving at Paris?
Possible, but unlikely. My passport has never been checked on that route (traveled there 3 or 4 times this past year).